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Viscoelastic boundary-layer flows past wedges and cones
ht. 1. Ennag Sri. Vol. $8. pp. 713-726 Q Pergamon Press Ltd.. 1980. Printed in Great Britain
VISCOELASTIC BOUNDARY-LAYER FLOWS PAST WEDGES AND CONES STEVE G. ROCHELLEt Tennessee Eastman Co., Kingsport,.TN 37662,U.S.A. and JOHN PEDDIESON, Jr.S Tennessee Technological University, Cookeville, TN 38501, U.S.A. Abstract-An implicit finite-difference method is used to solve the boundary-layer equations corresponding to the plane flow of a viscoelastic liquid past a symmetric wedge or the axisymmetric flow of such a fluid past a right circular cone. A representative sample of the computer data is displayed graphically and used to illustrate the interesting physical features of the problem.
INTRODUCTION C~NSTITUTIVE EQUATIONS which are able to describe the complex behavior of viscoeiastic liquids are necessarily complicated. At present the correct forms of such equations are irn~~e~tiy known. For these reasons many investigators have sought to avoid both much of the didn’t mathematics associated with the solution of realistic #low problems and the necessity for complete characterization of real viscoeiastic liquids by employing constitutive equations meant to describe slightly elastic fluids. A discussion of many of these approximate constitutive equations together with an extensive list of references is given by Bird[l]. In many cases the degree of accuracy inherent in such procedures is unknown. It would seem useful, therefore, to obtain accurate solutions to flow problems using relatively simple qualitatively correct constitutive equations so that these solutions can be used as standards against which to compare predictions based on the application of the slightly-elastic approximation to these same constitutive equations. The computation of some accurate solutions of the type mentioned above is the subject of this paper. The problem discussed in the present work is that of boundary-layer flow of a nonline~ Maxwell fluid past a symmetric wedge or a right circular cone. The appropriate boundary-layer equations are solved numerically and a variety of typical results are presented graphically in order to facilitate comparisons of the type previously mentioned. The Maxwell model was employed because it predicts qualitatively correct behavior over a wide range of shear rates in viscometric flows and because it has been applied to a variety of flow problems[2-51 without predicting results wildly at variance with physical experience. Boundary-layer problems were investigated because they are essentially l-dimensional flows where the shear rate varies in the primary flow direction as well as normal to it. They, therefore, represent one step up in complexity from viscometric flows where the shear rate varies only in the direction normal to the motion. There are infinitely many possible forms of the nonlinear Maxwell constitutive equation. The one used here was obtained by beginning with the general expression given by Eringen[6] and simplifying it as suggested by Oldroydf7J. For this reason the result is referred to as the Maxwell-Oidroyd constitutive equation in the present paper. It has the property that its predictions in viscometric flows are identical to those associated with the somewhat more complicated Oldroyd constitutive equation[7]. The only papers known to the present authors containing work directly pertinent to this investigation are those by Hsu[8] who found a similarity solution for the boundary-layer flow of an Oidroyd fluid past a 45” half-angle wedge, Rocheiie and Peddieson[9] who anaiysed the one-dimensional unsteady boundary-layer flow of a Maxwell-Oidroyd fluid induced by the ~En~neering M~hanicist. SAssociate Professor of En~neering Science and Mechanics. 713
714
S. G. ROCHELLE and J. PEDDIESON. JR
impulsive acceleration to uniform speed of an infinite flat plate, and Rochelle and Peddieson[ lo] who considered the steady boundary-layer flow of a Maxwell-Oldroyd fluid past a semi-infinite flat plate. Virtually all other boundary-layer solutions for viscoelastic fluids have been found using the slightly-elastic approximation. For a list of papers on this topic see the references given by Peddieson [ 111. It should be pointed out that it is not the purpose of this paper either to present a new numerical method or to exhibit new physical phenomena (although it is shown how certain types of behavior, already familiar from viscometer flow manifest themselves in boundary-layer flows). The Maxwell-Oldroyd constitutive equation was chosen precisely because its behavior is well understood and because, when combined with the balance laws and subjected to the order-of-magnitude analysis to be discussed subsequently, it leads to a system of field equations that can be solved by existing numerical methods. The purpose of the present work is to obtain solutions that can be used ‘as standards of comparison so as to obtain a good understanding of the ranges of validity of various perturbation methods which have been proposed to solve the equations arising when realistic (and therefore complicated) constitutive equations are used to analyse realistic flow problems. For a discussion of such asymptotic methods see Bird [ I] and the references contained therein. These asymptotic methods are the only systematic procedures that presently hold out the promise of obtaining useful predictions of the behavior of realistic viscoelastic flows. It is, therefore, felt that a good understanding of the accuracy of such methods is essential. Subsequently in this paper the most popular perturbation method, that based on the approximation of slight elasticity, is discussed in more detail. It is explained how the solutions obtained for the Maxwell-Oldroyd fluid can be used to test the validity of this approximation. These solutions could be used to test the accuracy of other asymptotic methods in a similar way. GOVERNING
EQUATIONS
Consider the axisymmetric flow of an incompressible viscoelastic liquid exhibiting mass density p and zero-shear-rate viscosity ii past a right circular cone of half-angle 0,. Let a characteristic length be denoted by L, a characteristic velocity by v, and a reciprocal Reynold’s number by e2 = i/(&v). Let the velocity vector be vv, the gradient operator be V/L, the time be Lt/v, and the stress tensor be pp$. The balance-of-mass equation then has the dimensionless form VT = 0.
(I)
Let $=-PJ+Eg
(2)
where P is the pressure, 11is the unit and 5 is the extra stress tensor. Then the balance-oflinear-momentum equation can be written in the dimensionless form Dtv = -VP + EV.~
(3)
Dtv = a,v + VTV
(4)
where
is the comoving derivative of the velocity vector. The Maxwell-Oldroyd constitutive equation employed in the present work can be written in dimensionless form as
(5) where <i/ v (i= 1,2, 3) are three material parameters having the dimensions of time, Q = (Vv + VVT)/2
(6)
715
Viscoelastic boundary-layer flows past wedges and cones
is the rate-of-strain tensor, and
is the convected derivative of the extra stress tensor. For flow past a cone it is convenient to employ a set of spherical polar coordinates (Lr, 8,~$) with associated unit vectors (e, eel e+). The coordinate system is positioned with the origin at the vertex of the cone and the cone surface at 13= 8,. The plane defined by e, and ee is a plane of symmetry for the flow. To derive the boundary-layer forms of the balance laws and constitutive equations for steady axisymmetric flow one substitutes V=
era,t e&Jr+ je+a,/(r sin 0) (i = 11,
(8)
v = e,v,( r, 6) + eeeve( r, e),
(9)
P = P(r,e),B= &8),
(10)
and 8=
ew +
(11)
&y
into (l), (3) and (5) and takes the limit as E approaches zero. This leads to the balance-of-mass equation
art+ t a,t+jr + (1 + j)vdr = 0,
(12)
the balance-of-linear-momentum equations vladb t vsayu,lr = - a,P t a,u&
a,P= 0, ayue4= 0,
(13)
and the constitutive equations a, - 2(7, + 72)ayu, ad/r = 273(ayudr)2 see - 2T2aYvr ad/r = -27,(a,udr)2 U~ - 7&u&r - (7, + 7,)a,vp& = a,vAr U,q,+ - (7, + 2r2)a,ufls.Jr - 72aYupeIr = 0 ur4 - (7,t 7Jayupe,Jr = 0, U~ = 0.
(14)
Equation (13b) shows that the pressure is constant across the boundary layer, while (13~) and (14 e,f) are satisfied by ur~=u&p=u~=o
(15)
The above equations can be put in a slightly more convenient form if tangential and normal coordinates (s,n) to the cone surface are used. The appropriate coordinate transformation is s = r cos(sy),
En = r sin(sy).
(16)
,The boundary-layer forms of (16) are found by taking the limit as E approaches zero to get s=
r, n = ry.
(17)
It can be shown that in the boundary-layer limit the velocity components transform according to the laws I.J.E.S. ISiS --E
716
S. G. ROCHELLE and J. PEDDJESON, JR &I = v, - nv,ls
vr = Us,
(18)
and the stress components transform according to a;, = ~sirwsee = %I,
rrpQ =
cr,,.
(19)
Substituting (17), (18) and (19) into (12), (13a) and (14a,b,c) yields the field equations
a,v, + a,~,+ ~v~/s = 0 v,&v, + a,a,a,Y= - P’ + ~~(~a~~*),P’ = dP(s)lds and the constitutive equations
(21) where El =
(1f x(~“vJ%(l+
(22)
y*@,v,P)
is the apparent viscosity. In (22) 7’1 = -
37, + 2Qh
‘y2 =
-4(n
(23)
+ 72h2.
In (20) P = P(s) is now interpreted, in the usual ~undary-layer manner, as the inviscid pressure at the wall of the cone. For plane flow past a wedge it can be shown that eqns (20)-(23) apply with j = 0. Defining the shear rate to be K =
a,v,
(24)
It can be seen that (21) and (22) can be rewritten as ass
=
2((7,
+
72)P
cr,,
=
/e&K, j.l =
(I+
-
73)KZ,
nK’)/(f
CT,,
+
=
y2K2).
2(?2p -
T3)K2
(25)
These equations have the forms appropriate to viscometric flow with shear rate K [12]. For viscometric flows K is a function of only one variable. In the present case K depends on two variables. If this is called a locally viscometric state of stress, it can be seen that the order-of-magnitude analysis used to derive the boun~ry-layer equations in this paper is the one which leads to a locally viscometric state of stress. This is the same order-of-magnitude analysis used by Hsu[8]. There is no unique boundary-layer theory for a Maxwell-Oldroyd fluid. Different ways of ordering the three time constants appearing in the constitutive equations will produce different sets of boundary-layer equations. The locally viscometric boundary-layer equations are used in the present work because it is felt that their solutions can be used conveniently to investigate the questions discussed in the introduction. Inspection of (20) reveals that the only non-Newtonian effect present in the field equations is that of variable viscosity. Thus the velocity components associated with the locally viscometric boundary-layer theory could be computed by assuming that the fluid was a purely viscous material obeying the apparent viscosity relationship associated with viscometric flow. Such an approach would be sufficient to compute the skin friction but, of course, would yield no infor~tion about the normat stress effects. These will be discussed subsequently.
Viscoelasticboundary-layerflowspast wedgesand cones
717
To solve (20) ~undary conditions are needed. These are v,(qO) = 0,
u,(s,O) = 0, lim U&I) = sp. “-00
(26)
Equations (26a,b) come from the no-slip condition while (26~) is the usual boundary-layer condition. The ~ght-h~d side of (26~) represents the expression for the inviscid surface speed which results from identifying the characteristic velocity F with the inviscid surface speed at a distance l from the vertex of the wedge or cone. The value of p is related to the value of 0, [ 131and obeys the inequality 0 I p 5 1. To facilitate the numerical solution of the equations it is convenient to make the following transformations of variables suggested by the Falkner-Skan variablesll3j.
matching
s = [/( 1 - &), n = (2/(1 -I-P))“*f@(I - #8-p%j V” = (2/( 1 + p))‘Q/(l - .#j+‘)‘2 % = (541 - w m,$, ~((1 --~)rl F(Z9$/2+ (i+ (1+~)/2) G&r))).
(27)
substituti~ (27) into (201, (21), (22) and (26) yields a,,G+F+(2/(1+2j+p)) z‘(l-&+F=O a,(~a,F)-(l+2il(lt~))Ga,F+(2~/(l+p))(l-F2) - (2/( 1+ P)) &I - W&F = 0 F(&O)= G(&O)= 0, lim F(&) P 1 ?-
(28) (2%
Equations (28) are parabolic and are thus amenable to solution by numerical methods developed for Newtonian boundary-layer problems. It is interesting to note that these are the correct locally viscometric Sunday-layer equations for flow of any viscoeiastic liquid past a wedge or a cone. Only the form of p disti~ishes one such liquid from another. RESULTS AND DISCUSSION
The complicated form of (28) makes the possibility of closed-form solutions highly unlikely, especially in view of the fact that none have yet been found in the Newtonian case (g = 1). For this reason the govemi~ equations were solved nu~~ca~y by an implicit fi~te~~erence method. (See Blottnerllrl] for a discussion of the application of such methods to Newtonian boundary-layer problems.) The procedure employed in the present work can be briefly described as follows. The [ derivatives in (28) were represented by two-point backward difference quotients. This created a set of two ordinary differential equations at each value of 4 which could be solved if the solution at the previous value of 5 was known. At 5 = 0 the (4derivatives in (28) vanish. This allows the solution process to start at 5 = 0 and march forward in the 5 direction. At a given value of .$ the ordinary differential equations mentioned above were solved iteratively (due to the nonlinearity in (28b)) by applying the method of quasilinearization discussed by Bellman and Kalaba[lS] to (28b). Given F from the previous iteration (2%~)was integrated by the trapezoidal rule starting at v = 0 to determine G at all values of q. Then the derivatives in the linearized form of (28b) produced by q~siline~tion were represented by the three-point central difference quotients. This when combined with @a) and (29~)~resulted in a &i-diagonal set of linear algebraic equations which was solved (with the G’s being known from the integration of (28a)) for the values of F at all r)‘s. Since the limit indicated in (29~) is asymptotic,
718
S. G. ROCHELLE and J. PEDDIESON, JR
it was necessary to integrate out only to about 1)= 7 to achieve satisfaction of this condition. In fact all integrations were carried out to approximately 77= 10. The [ step size was held constant in the procedure. The q step size was allowed to vary in such a way that the ratio of any two successive step sizes was a constant slightly greater than unity. This allowed the use of very small step sizes near the body surface and produced a situation in which the 7 step sizes slowly increased with increasing 7. The use of small step sizes near n = 0 was found to be necessary in many of the situations encountered. Before proceeding to a discussion of the numerical results it is appropriate to mention some of the properties of the Maxwell-Oldroyd model. It can be shown[l6] that, in order for the shear stress in viscometric flow to be an increasing function of the shear rate, the inequalities l/9 5 y,/y* 5 1
(32)
must be satisfied. From (25d) it can be seen that (in view of (32)) b 5 1 and lii p = 1, lim p = yl/y2. K-t=
(33)
Thus the apparent viscosity is a decreasing function of the shear rate. The apparent viscosity at infinite shear rate (the minimum value) is one ninth of the zero-shear-value. The Maxwell-Oldroyd model can, therefore, describe only shear-thinnmg liquids and exhibits a minimum apparent viscosity. Since many real viscoelastic liquids are known[l] to have infinite-shear-rate apparent viscosities which are less than one ninth of their zero-shear-rate viscosities, this constitutes an important restriction on the applicability of this constitutive equation. In order for the model to exhibit creep a second inequality must be satisfied. It is 71
h
0.
(34)
It is convenient to regard TV,yl and y2 as the fundamental material constants. Equations (23) can then be solved for 72 and ~3to obtain 72 =
-
73 =
71(1+_(1 - 72/$“2)/2, f yJ(271(1 - &W).
(35)
It can be shown that in order for the normal-stress effects to have the signs indicated by the preponderance of experimental evidence the sign ambiguities in (35) must be resolved in favor of the minus signs. Thus the equations 72= - T,(1 - (1 - &:)“2)/2, 5
=
-
y1/m1u
-
(36)
72W2)
were used in this paper. An obvious consequence of (36) is the inequality y2
5
7:.
(37)
In the locally viscometric boundary-layer problems under discussion here the shear rate is K = ((1 + p)/2)“2(& 1 - .!J))(3p-%&
(38)
Thus for p < l/3 the shear rate decreases from intinity to zero as ,$ goes from zero to one, for p > l/3 th&ear rate increases from zero to infinity as [goes from zero to one, while forp = l/3 the shear rate is in&l%ident of 6. From (30) it can be seen that
1 Yh2
p(o,~) =
(1 + 2~0,~)2/3)1(I+ 1
~Y~@,F)‘/G
p<1/3 , P = l/3 , P>1/3
(39)
719
Viscoelastic boudnary-layer flows past wedges and cones
p
1 di,d= 1
(1+2n(a,~)*/3)1(1+2rz(d,~)2/3j, P = 113 p > l/3 9 YJY2
WI
Equations (39) and (40) are indicative of the fact that the solutions to be discussed exhibit a transition between low-shear-rate and high-shear-rate behavior. For p < l/3 the low-shear-rate region is far down stream from the vertex while for p > l/3 it is in the immediate vicinity of the vertex. For p = l/3 (28) is satisfied by F = F(q). Thus the shear rate (and, therefore, the apparent viscosity) is independent of .$ It is this case, which corresponds to a wedge or cone with a 45” half-angle, that was considered by Hsu[8] because it admitted a similarity solution. In carrying out the numerical calculations the parametric values were chosen so as to illustrate parametric trends. No attempt was made to represent any specific material. For simplicity the value of y2 was fixed at unity. In accord with (34) and (37) 71 was fixed at two (this affects only the normal stresses). To be consistent with (32), values of yI between 0.15 and unity were employed. As indicated earlier, the prime motivation for this work was to provide exact results that could be used to evaluate the accuracy of proposed approximate methods of solving viscoelastic flow problems. For this reason a variety of results are presented for both wedges and cones. Velocity profiles were computed but these are not reported for the sake of brevity. Also, no results are given for p = 0 (flat plate and thin needle) because these have been reported elsewhere[lO] for the flat plate and because the effect of transverse curvature, important in the thin needle problem, has been neglected in the present work. Figures 1 and 2 contain typical results for the wall apparent viscosity function PI%’ = P(5,O).
(41)
The variation of the viscosity with wall position is as indicated by (39) and (40). The curves exhibit increasingly large gradients as p = l/3 is approached from either side. For a Newtonian fluid yl = y2 = 0 which implies that or.= 1. For p = 1 (23) is satisfied by F = F(q) and G = G(q). Thus for a Newtonian fluid similarity solutions exist for all the cases under discussion here. The Newtonian value of any of the quantities plotted in these or subsequent graphs is identical to the zero-shear-rate limit of that quantity. These and all subsequent graphs were drawn by an x*y plotter interfaced with a Xerox Sigma 6 computer. The slope discontinuities in some of the curves are due to the equal spacing of points used by the plotter and not to any discontinuities in the computed solutions. For the wedge the vertex half-angle is related to p by the formula elv= TPl(l -PI.
0.2
0.4 0.6 Fig. 1. Apparent viscosity for wedge.
(42)
0.8
1.0 E
S.G.ROCHELLE and J.PEDDIESON,JR
720
0.2
0,o
0.4
0.6
0.8
E
1.0
Fig. 2. Apparent viscosity for cone.
For the cone the co~es~nding reIationsh~p is given by Rosenhead[l3]. A few representative values are as follows. Half-angles of 30“, 45”, W, and 90” correspond, respectively, to p = 0.20, l/3,0.50 and 1.0 for the wedge and to p = 0.12, l/3,0.42 and 1.0 for the cone. It is interesting to note that a similarity solution does not exist for either plane or axisymmetric stagnation-point flow 0, = 1) when the Maxwell-Oldroyd model is used. Some typical results for the shear-stress (skin-friction) coefficient
are shown in Figs. 3 and 4. It can be seen that in all cases the maximum and minimum values of the skin-friction coefficient lie at the end points of the region and that the viscoelastic skin-friction coefficient is always less than the co~espondi~ Newtonian value (often substantially less for these values of yr and ~2). This is in contrast to the case of the flat piate[lO] where an overshoot phenomenon is observed and the maximum value of the skin-friction coefficient (which is larger than the Newtonian value) occurs in the middle of the region. In observing these and subsequent figures it is important to recall that the physical distance from
0.8
0,4
0.2
Fig. 3. Shear
0.6
stress coefficient for wedge.
ha
t
1.0
721
Viscoelastic budnary-layer flows past wedges and cones
C
0.3 ’ 0.0
0.2
0.4
006
0.8 E
1.0
Fig. 4. Shear stress coefficient for cone.
the vertex is severely compressed by the use of the variable 6 (rather than s) especially for large values of s (value of 5 near unity). Thus large gradients with respect to 5 do not necessarily reflect large changes as can be seen from the relationship
where A is any variable. The variable 5 is employed only because it simplifies the numerical solution and the presentation of the data. Data for the normal-stress coefficients c,, = (2/(1 +p))W CWI
=
(2/U
+ PW1
- 5))(’- %,,(&O) = 2((r, + 72)/45,0) = 2(T*P(&O) - 5)) (’ - 3%,,(&0)
-
T3)@,ww2
(45)
73)(4mm*
for the wedges only are shown in Figs. 5 and 6. To see the significance of these quantities it is necessary to refer to (2) which, when combined with (45), yields
0.3 0.2 2.c
0.0
0.1
0.2
0.4
0.6
Fig. 5. First normal stress coeficient for wedge.
0.8
E
1.0
S. C. ROCHELLE
722
and J. PEDDlESON
JR
0.5
C nn 0.4
0.3
0,2
0.1
0.0 0 0.1 -’ Fig. 6. Second normal stress coefficient
for wedge.
&9(&O)= -(Po - R/(1 - ‘9)2”(1- F(1 + p)([/(l - %9)‘“-“c,,)/2) Ll(5.0) = -(Po- (Y(1 - 5))2p(l - &(I +p)(&l - 5))‘p-‘)Cn.)/2)
(4)
where p. is the pressure at the vertex. For a Newtonian fluid C,, = C,,, = 0. It can be seen that there is a limited region near [ = 0 where the extra normal stresses, even though multiplied by E, have a magnitude equal to or exceeding that of the pressure. It should be noted, however, that the fundamental assumption of boundary-layer theory namely a, * &a,, is violated near 5 = 0. Thus boundary-layer theory cannot be applied with confidence in this region. Away from ,$= 0, where the boundary-layer equations are applicable, the elastic normal-stress effects are small compared to the contribution of the pressure. In view of these facts it would appear that the C,, and C,, play a relatively minor role in determining the forces applied to a wedge or cone by a viscoelastic liquid. For brevity, therefore, results are shown for the wedge only. It should be recalled that the normal stress perpendicular to the s,n plane is predicted to be zero by the Maxwell-Oldroyd constitutive equation. Thus ~~~-a,,, and u”, respectively are the quantities generally referred to as the first and second normal stress differences. It can be seen that C,,-C,,, (and thus ~~,-a,,) are predicted to be positive at all wall positions. This result is in agreement with observed behavior in viscometric flows. It can be further seen that C,. (and thus Us,,) is predicted to be positive in the high-shear-rate limit and negative in the low-shear-rate limit. Experimental evidence is not conclusive on this point, with some experimenters finding the second normal stress difference to be positive in viscometric flows while others observe it to be negative. The present results suggest that the sign may be a function of the shear rate. The displacement thickness for wedge and cone flows is defined to be S=
I
a
(1 - v,Jsp) dn.
0
(47)
Figures 7 and 8 display results for the displacement-thickness coefficient C, = (( 1+ p)/2)“*(51(1 - [))‘p-1)‘2S= [( 1 - F) dn.
(48)
It can be observed that this quantity is always decreased by fluid elasticity and that for the values of yI and y2 shown the decrease can be substantial. Figures l-8 all contain results for y = 0.4. As yl approaches y2 (fixed at unity in these calculations) the apparent viscosity approaches a constant value of unity. Since the variable viscosity is the only non-Newtonian effect present in the field eqns (28), the solutions for F and G will be the same as for a Newtonian fluid when yl = y2. It should be pointed out, however, that the normal stresses do not vanish when yl = y2 unless both are equal to zero. Thus it is not
Viscoelastic
boudnary-layer
723
flows past wedges and cones
0.8
O-6
0.5
0.4 0.0
0.2
0.4 Fig. 7. Displacment
0.6
0.8
c
1.0
thickness fur wedge.
cl 3 so,4
r,=l,O
0.6
o,o
0.2
0.4 Fig. 8. Displacement
0.6 thickness for cone.
0.8
E
1
724
S. G. ROCHELLE and 1. PEDDIESON, JR 1.0
C
x = 1.0
.n
0.8
0,6
0.6
OS2
WEDGE
o,oI
I
I
I
.
*
.
I
.
0*5
0.0
.
,
E
o,ol
1,o
CONE
.
.
*
,
,
0.0
,
,
,
,
OS5
,
6
1.0
Fig. 9. Shear stress coefficient for wedge and cone.
strictly correct to refer to the situation where y1 = y2 as the Newtonian case. This terminology, however, will be used in the present discussion because, as previously stated, the normal stress effects are of secondary importance in locally viscometric boundary-layer problems. Figures 9 and 10 use the skin-friction coefficient to illustrate the amount of deviation from Newtonian behavior associated with various values of yl. It can be seen that the results for wedges and cones are qualitatively similar with about the same order of magnitude of skin-friction reduction exhibited for a given value of yI. Since the wedge and cone angles associated with a given value of p are different, a fe\N results for the skin-friction coefficient are presented in Fig. 11 for selected vertex angles. For all angles shown the results are qualitatively similar. Since constitutive equations intended to describe the behavior of real viscoelastic liquids are
ls2 C ‘”
1.2 C .” 1.0
088
L-J r,=l.O w -60”
0.6
O.l;\\
0,6
WEDGE
0.4
i 0.4
0,2 0.0
0.5
E
4 0.2 1.0 0.0
CONE
OS5
Fig. 10. Shear stress coefficient for wedge and cone.
E
1,1)
Viscoelastic boundary-layer flows past wedges and cones
0.8
0.8
086
006
12s
Fig. 11.Shear stress coefficient for wedge and cone.
highly complex it has often been proposed that approximate solutions be obtained either by perturbation methods or by using approximations to the general constitutive equations that are valid for small elastic effects. It is hoped that the solutions obtained in the present work can be used to test some of these proposed approximation methods in a specific case. Even without the aid of explicit calculations it is possible to observe that it will be easire to obtain solutions for small elastic effects when p > l/3 than when p < l/3. For the locally viscometric boundary-layer theory the approximation of small elastic effects would amount to using an approximate form of CL,found by expanding (30) for 714 1 and y2 Q 1, in (28). This would not change the parabolic nature of (28) so the solution would still have to start at ,$= 0 and proceed in the direction of increasing 6 For p > l/3 the elastic effects are smallest at 5 = 0. Thus the only question would be how close one could get to 5 = 1 before the approximation would break down. For p < l/3, on the other hand, the elastic effects are largest near ,$= 0 and expansion of (30) for y1 Q 1 and y2 & 1 would be invalid in this region because these parameters are multiplied by functions of 5 which become infinite at 5 = 0. This makes it doubtful that an approximate solution could be obtained because the initial conditions would have to be applied in a region where the approximation is invalid. If the initial conditions are not applied unavoidable ambiguities appear in the solution as indicated by the work of Davis[17] using the second-order constitutive equation. It may be possible to resolve the problem discussed above by formulating another approximate constitutive equation valid for large elastic effects and using it as a basis for solutions when p < l/3. It does not appear that much attention has yet been devoted to equations of this type. CONCLUSION
In this paper the locally viscometric boundary-layer equations associated with a MaxwellOldroyd viscoelastic fluid have been solved numerically by an implicit finite-difference method for flow of such a fluid past a wedge or a circular cone. Selected numerical results were presented graphically and the important aspects of these were discussed. REFERENCES [l] R. B. BIRD, Ann. Rev. Fluid Mech. 8, 13 (1976). [2] R. ROTHENBERGER, D. H. MCCOYand M. M. DENN, Trans. .Soc. Rheol. 17,259 (1973). [3] 2. 4. SUN and M. M. DENN, A.Z.C’.h.E.J. 18, 1010(1972). 141M. M. DENN, C. J. S. PETRIE and P. AVENAS, A.1Ch.E .l 21,791 (1975). [S] R. J. FISHER and M. M. DENN, A.Z.C.h.E .I. 23.23 (1977). 161A. C. ERINGEN, Mechanics of Conlinus, Chap. 9. Wiley, New York (1%7).
726
S. G. ROCHELLE and J. PEDDIESON. JR
[7] J. G. OLDROYD, Proc. Roy Sot. Lond. 245A. 278 (19.58). [8] C. -C. HSU, J. Fluid Mech. 27,445 (1967). [9] S. G. ROCHELLE and 1. PEDDIESON, Proc. Twelfth Annual Meeting Sot. Engng Sci. 1159 (1975). [IO] S. G. ROCHELLE and J. PEDDIESON, Proc. Twelfth Southeastern Seminar on Thermal Sciences 113 (1976). [li] J. PEDDIESON, Deu. Mech. 6, 153 (1971). [12] A. G. FREDRICKSON, Principles and Applications of Rheology. Chap. 7. Prentice-Hall. En&wood Cliffs (1964). [13] L. ROSENHEAD (editor), Laminar Boundary Layers. Chap. 5. Oxford University Press. London (1%3). [14] F. G. BLO’ITNER, AIAA J. 8. 193 (1970). [IS] R. E. BELLMAN and R. E. KALABA, Quasilinearization and Nonlinear Boundary-Value Problems, Chap. 2. American Elsevier, New York (1%5). 1161J. G. OLDROYD, Rheol. Acta I, 337 (]%I). [17] R. T. DAVIS, Deu. Mech. 4, 1145(1%7).
(Received
for publication
8 Juiy 1979)
VISCOELASTIC BOUNDARY-LAYER FLOWS PAST WEDGES AND CONES STEVE G. ROCHELLEt Tennessee Eastman Co., Kingsport,.TN 37662,U.S.A. and JOHN PEDDIESON, Jr.S Tennessee Technological University, Cookeville, TN 38501, U.S.A. Abstract-An implicit finite-difference method is used to solve the boundary-layer equations corresponding to the plane flow of a viscoelastic liquid past a symmetric wedge or the axisymmetric flow of such a fluid past a right circular cone. A representative sample of the computer data is displayed graphically and used to illustrate the interesting physical features of the problem.
INTRODUCTION C~NSTITUTIVE EQUATIONS which are able to describe the complex behavior of viscoeiastic liquids are necessarily complicated. At present the correct forms of such equations are irn~~e~tiy known. For these reasons many investigators have sought to avoid both much of the didn’t mathematics associated with the solution of realistic #low problems and the necessity for complete characterization of real viscoeiastic liquids by employing constitutive equations meant to describe slightly elastic fluids. A discussion of many of these approximate constitutive equations together with an extensive list of references is given by Bird[l]. In many cases the degree of accuracy inherent in such procedures is unknown. It would seem useful, therefore, to obtain accurate solutions to flow problems using relatively simple qualitatively correct constitutive equations so that these solutions can be used as standards against which to compare predictions based on the application of the slightly-elastic approximation to these same constitutive equations. The computation of some accurate solutions of the type mentioned above is the subject of this paper. The problem discussed in the present work is that of boundary-layer flow of a nonline~ Maxwell fluid past a symmetric wedge or a right circular cone. The appropriate boundary-layer equations are solved numerically and a variety of typical results are presented graphically in order to facilitate comparisons of the type previously mentioned. The Maxwell model was employed because it predicts qualitatively correct behavior over a wide range of shear rates in viscometric flows and because it has been applied to a variety of flow problems[2-51 without predicting results wildly at variance with physical experience. Boundary-layer problems were investigated because they are essentially l-dimensional flows where the shear rate varies in the primary flow direction as well as normal to it. They, therefore, represent one step up in complexity from viscometric flows where the shear rate varies only in the direction normal to the motion. There are infinitely many possible forms of the nonlinear Maxwell constitutive equation. The one used here was obtained by beginning with the general expression given by Eringen[6] and simplifying it as suggested by Oldroydf7J. For this reason the result is referred to as the Maxwell-Oidroyd constitutive equation in the present paper. It has the property that its predictions in viscometric flows are identical to those associated with the somewhat more complicated Oldroyd constitutive equation[7]. The only papers known to the present authors containing work directly pertinent to this investigation are those by Hsu[8] who found a similarity solution for the boundary-layer flow of an Oidroyd fluid past a 45” half-angle wedge, Rocheiie and Peddieson[9] who anaiysed the one-dimensional unsteady boundary-layer flow of a Maxwell-Oidroyd fluid induced by the ~En~neering M~hanicist. SAssociate Professor of En~neering Science and Mechanics. 713
714
S. G. ROCHELLE and J. PEDDIESON. JR
impulsive acceleration to uniform speed of an infinite flat plate, and Rochelle and Peddieson[ lo] who considered the steady boundary-layer flow of a Maxwell-Oldroyd fluid past a semi-infinite flat plate. Virtually all other boundary-layer solutions for viscoelastic fluids have been found using the slightly-elastic approximation. For a list of papers on this topic see the references given by Peddieson [ 111. It should be pointed out that it is not the purpose of this paper either to present a new numerical method or to exhibit new physical phenomena (although it is shown how certain types of behavior, already familiar from viscometer flow manifest themselves in boundary-layer flows). The Maxwell-Oldroyd constitutive equation was chosen precisely because its behavior is well understood and because, when combined with the balance laws and subjected to the order-of-magnitude analysis to be discussed subsequently, it leads to a system of field equations that can be solved by existing numerical methods. The purpose of the present work is to obtain solutions that can be used ‘as standards of comparison so as to obtain a good understanding of the ranges of validity of various perturbation methods which have been proposed to solve the equations arising when realistic (and therefore complicated) constitutive equations are used to analyse realistic flow problems. For a discussion of such asymptotic methods see Bird [ I] and the references contained therein. These asymptotic methods are the only systematic procedures that presently hold out the promise of obtaining useful predictions of the behavior of realistic viscoelastic flows. It is, therefore, felt that a good understanding of the accuracy of such methods is essential. Subsequently in this paper the most popular perturbation method, that based on the approximation of slight elasticity, is discussed in more detail. It is explained how the solutions obtained for the Maxwell-Oldroyd fluid can be used to test the validity of this approximation. These solutions could be used to test the accuracy of other asymptotic methods in a similar way. GOVERNING
EQUATIONS
Consider the axisymmetric flow of an incompressible viscoelastic liquid exhibiting mass density p and zero-shear-rate viscosity ii past a right circular cone of half-angle 0,. Let a characteristic length be denoted by L, a characteristic velocity by v, and a reciprocal Reynold’s number by e2 = i/(&v). Let the velocity vector be vv, the gradient operator be V/L, the time be Lt/v, and the stress tensor be pp$. The balance-of-mass equation then has the dimensionless form VT = 0.
(I)
Let $=-PJ+Eg
(2)
where P is the pressure, 11is the unit and 5 is the extra stress tensor. Then the balance-oflinear-momentum equation can be written in the dimensionless form Dtv = -VP + EV.~
(3)
Dtv = a,v + VTV
(4)
where
is the comoving derivative of the velocity vector. The Maxwell-Oldroyd constitutive equation employed in the present work can be written in dimensionless form as
(5) where <i/ v (i= 1,2, 3) are three material parameters having the dimensions of time, Q = (Vv + VVT)/2
(6)
715
Viscoelastic boundary-layer flows past wedges and cones
is the rate-of-strain tensor, and
is the convected derivative of the extra stress tensor. For flow past a cone it is convenient to employ a set of spherical polar coordinates (Lr, 8,~$) with associated unit vectors (e, eel e+). The coordinate system is positioned with the origin at the vertex of the cone and the cone surface at 13= 8,. The plane defined by e, and ee is a plane of symmetry for the flow. To derive the boundary-layer forms of the balance laws and constitutive equations for steady axisymmetric flow one substitutes V=
era,t e&Jr+ je+a,/(r sin 0) (i = 11,
(8)
v = e,v,( r, 6) + eeeve( r, e),
(9)
P = P(r,e),B= &8),
(10)
and 8=
ew +
(11)
&y
into (l), (3) and (5) and takes the limit as E approaches zero. This leads to the balance-of-mass equation
art+ t a,t+jr + (1 + j)vdr = 0,
(12)
the balance-of-linear-momentum equations vladb t vsayu,lr = - a,P t a,u&
a,P= 0, ayue4= 0,
(13)
and the constitutive equations a, - 2(7, + 72)ayu, ad/r = 273(ayudr)2 see - 2T2aYvr ad/r = -27,(a,udr)2 U~ - 7&u&r - (7, + 7,)a,vp& = a,vAr U,q,+ - (7, + 2r2)a,ufls.Jr - 72aYupeIr = 0 ur4 - (7,t 7Jayupe,Jr = 0, U~ = 0.
(14)
Equation (13b) shows that the pressure is constant across the boundary layer, while (13~) and (14 e,f) are satisfied by ur~=u&p=u~=o
(15)
The above equations can be put in a slightly more convenient form if tangential and normal coordinates (s,n) to the cone surface are used. The appropriate coordinate transformation is s = r cos(sy),
En = r sin(sy).
(16)
,The boundary-layer forms of (16) are found by taking the limit as E approaches zero to get s=
r, n = ry.
(17)
It can be shown that in the boundary-layer limit the velocity components transform according to the laws I.J.E.S. ISiS --E
716
S. G. ROCHELLE and J. PEDDJESON, JR &I = v, - nv,ls
vr = Us,
(18)
and the stress components transform according to a;, = ~sirwsee = %I,
rrpQ =
cr,,.
(19)
Substituting (17), (18) and (19) into (12), (13a) and (14a,b,c) yields the field equations
a,v, + a,~,+ ~v~/s = 0 v,&v, + a,a,a,Y= - P’ + ~~(~a~~*),P’ = dP(s)lds and the constitutive equations
(21) where El =
(1f x(~“vJ%(l+
(22)
y*@,v,P)
is the apparent viscosity. In (22) 7’1 = -
37, + 2Qh
‘y2 =
-4(n
(23)
+ 72h2.
In (20) P = P(s) is now interpreted, in the usual ~undary-layer manner, as the inviscid pressure at the wall of the cone. For plane flow past a wedge it can be shown that eqns (20)-(23) apply with j = 0. Defining the shear rate to be K =
a,v,
(24)
It can be seen that (21) and (22) can be rewritten as ass
=
2((7,
+
72)P
cr,,
=
/e&K, j.l =
(I+
-
73)KZ,
nK’)/(f
CT,,
+
=
y2K2).
2(?2p -
T3)K2
(25)
These equations have the forms appropriate to viscometric flow with shear rate K [12]. For viscometric flows K is a function of only one variable. In the present case K depends on two variables. If this is called a locally viscometric state of stress, it can be seen that the order-of-magnitude analysis used to derive the boun~ry-layer equations in this paper is the one which leads to a locally viscometric state of stress. This is the same order-of-magnitude analysis used by Hsu[8]. There is no unique boundary-layer theory for a Maxwell-Oldroyd fluid. Different ways of ordering the three time constants appearing in the constitutive equations will produce different sets of boundary-layer equations. The locally viscometric boundary-layer equations are used in the present work because it is felt that their solutions can be used conveniently to investigate the questions discussed in the introduction. Inspection of (20) reveals that the only non-Newtonian effect present in the field equations is that of variable viscosity. Thus the velocity components associated with the locally viscometric boundary-layer theory could be computed by assuming that the fluid was a purely viscous material obeying the apparent viscosity relationship associated with viscometric flow. Such an approach would be sufficient to compute the skin friction but, of course, would yield no infor~tion about the normat stress effects. These will be discussed subsequently.
Viscoelasticboundary-layerflowspast wedgesand cones
717
To solve (20) ~undary conditions are needed. These are v,(qO) = 0,
u,(s,O) = 0, lim U&I) = sp. “-00
(26)
Equations (26a,b) come from the no-slip condition while (26~) is the usual boundary-layer condition. The ~ght-h~d side of (26~) represents the expression for the inviscid surface speed which results from identifying the characteristic velocity F with the inviscid surface speed at a distance l from the vertex of the wedge or cone. The value of p is related to the value of 0, [ 131and obeys the inequality 0 I p 5 1. To facilitate the numerical solution of the equations it is convenient to make the following transformations of variables suggested by the Falkner-Skan variablesll3j.
matching
s = [/( 1 - &), n = (2/(1 -I-P))“*f@(I - #8-p%j V” = (2/( 1 + p))‘Q/(l - .#j+‘)‘2 % = (541 - w m,$, ~((1 --~)rl F(Z9$/2+ (i+ (1+~)/2) G&r))).
(27)
substituti~ (27) into (201, (21), (22) and (26) yields a,,G+F+(2/(1+2j+p)) z‘(l-&+F=O a,(~a,F)-(l+2il(lt~))Ga,F+(2~/(l+p))(l-F2) - (2/( 1+ P)) &I - W&F = 0 F(&O)= G(&O)= 0, lim F(&) P 1 ?-
(28) (2%
Equations (28) are parabolic and are thus amenable to solution by numerical methods developed for Newtonian boundary-layer problems. It is interesting to note that these are the correct locally viscometric Sunday-layer equations for flow of any viscoeiastic liquid past a wedge or a cone. Only the form of p disti~ishes one such liquid from another. RESULTS AND DISCUSSION
The complicated form of (28) makes the possibility of closed-form solutions highly unlikely, especially in view of the fact that none have yet been found in the Newtonian case (g = 1). For this reason the govemi~ equations were solved nu~~ca~y by an implicit fi~te~~erence method. (See Blottnerllrl] for a discussion of the application of such methods to Newtonian boundary-layer problems.) The procedure employed in the present work can be briefly described as follows. The [ derivatives in (28) were represented by two-point backward difference quotients. This created a set of two ordinary differential equations at each value of 4 which could be solved if the solution at the previous value of 5 was known. At 5 = 0 the (4derivatives in (28) vanish. This allows the solution process to start at 5 = 0 and march forward in the 5 direction. At a given value of .$ the ordinary differential equations mentioned above were solved iteratively (due to the nonlinearity in (28b)) by applying the method of quasilinearization discussed by Bellman and Kalaba[lS] to (28b). Given F from the previous iteration (2%~)was integrated by the trapezoidal rule starting at v = 0 to determine G at all values of q. Then the derivatives in the linearized form of (28b) produced by q~siline~tion were represented by the three-point central difference quotients. This when combined with @a) and (29~)~resulted in a &i-diagonal set of linear algebraic equations which was solved (with the G’s being known from the integration of (28a)) for the values of F at all r)‘s. Since the limit indicated in (29~) is asymptotic,
718
S. G. ROCHELLE and J. PEDDIESON, JR
it was necessary to integrate out only to about 1)= 7 to achieve satisfaction of this condition. In fact all integrations were carried out to approximately 77= 10. The [ step size was held constant in the procedure. The q step size was allowed to vary in such a way that the ratio of any two successive step sizes was a constant slightly greater than unity. This allowed the use of very small step sizes near the body surface and produced a situation in which the 7 step sizes slowly increased with increasing 7. The use of small step sizes near n = 0 was found to be necessary in many of the situations encountered. Before proceeding to a discussion of the numerical results it is appropriate to mention some of the properties of the Maxwell-Oldroyd model. It can be shown[l6] that, in order for the shear stress in viscometric flow to be an increasing function of the shear rate, the inequalities l/9 5 y,/y* 5 1
(32)
must be satisfied. From (25d) it can be seen that (in view of (32)) b 5 1 and lii p = 1, lim p = yl/y2. K-t=
(33)
Thus the apparent viscosity is a decreasing function of the shear rate. The apparent viscosity at infinite shear rate (the minimum value) is one ninth of the zero-shear-value. The Maxwell-Oldroyd model can, therefore, describe only shear-thinnmg liquids and exhibits a minimum apparent viscosity. Since many real viscoelastic liquids are known[l] to have infinite-shear-rate apparent viscosities which are less than one ninth of their zero-shear-rate viscosities, this constitutes an important restriction on the applicability of this constitutive equation. In order for the model to exhibit creep a second inequality must be satisfied. It is 71
h
0.
(34)
It is convenient to regard TV,yl and y2 as the fundamental material constants. Equations (23) can then be solved for 72 and ~3to obtain 72 =
-
73 =
71(1+_(1 - 72/$“2)/2, f yJ(271(1 - &W).
(35)
It can be shown that in order for the normal-stress effects to have the signs indicated by the preponderance of experimental evidence the sign ambiguities in (35) must be resolved in favor of the minus signs. Thus the equations 72= - T,(1 - (1 - &:)“2)/2, 5
=
-
y1/m1u
-
(36)
72W2)
were used in this paper. An obvious consequence of (36) is the inequality y2
5
7:.
(37)
In the locally viscometric boundary-layer problems under discussion here the shear rate is K = ((1 + p)/2)“2(& 1 - .!J))(3p-%&
(38)
Thus for p < l/3 the shear rate decreases from intinity to zero as ,$ goes from zero to one, for p > l/3 th&ear rate increases from zero to infinity as [goes from zero to one, while forp = l/3 the shear rate is in&l%ident of 6. From (30) it can be seen that
1 Yh2
p(o,~) =
(1 + 2~0,~)2/3)1(I+ 1
~Y~@,F)‘/G
p<1/3 , P = l/3 , P>1/3
(39)
719
Viscoelastic boudnary-layer flows past wedges and cones
p
1 di,d= 1
(1+2n(a,~)*/3)1(1+2rz(d,~)2/3j, P = 113 p > l/3 9 YJY2
WI
Equations (39) and (40) are indicative of the fact that the solutions to be discussed exhibit a transition between low-shear-rate and high-shear-rate behavior. For p < l/3 the low-shear-rate region is far down stream from the vertex while for p > l/3 it is in the immediate vicinity of the vertex. For p = l/3 (28) is satisfied by F = F(q). Thus the shear rate (and, therefore, the apparent viscosity) is independent of .$ It is this case, which corresponds to a wedge or cone with a 45” half-angle, that was considered by Hsu[8] because it admitted a similarity solution. In carrying out the numerical calculations the parametric values were chosen so as to illustrate parametric trends. No attempt was made to represent any specific material. For simplicity the value of y2 was fixed at unity. In accord with (34) and (37) 71 was fixed at two (this affects only the normal stresses). To be consistent with (32), values of yI between 0.15 and unity were employed. As indicated earlier, the prime motivation for this work was to provide exact results that could be used to evaluate the accuracy of proposed approximate methods of solving viscoelastic flow problems. For this reason a variety of results are presented for both wedges and cones. Velocity profiles were computed but these are not reported for the sake of brevity. Also, no results are given for p = 0 (flat plate and thin needle) because these have been reported elsewhere[lO] for the flat plate and because the effect of transverse curvature, important in the thin needle problem, has been neglected in the present work. Figures 1 and 2 contain typical results for the wall apparent viscosity function PI%’ = P(5,O).
(41)
The variation of the viscosity with wall position is as indicated by (39) and (40). The curves exhibit increasingly large gradients as p = l/3 is approached from either side. For a Newtonian fluid yl = y2 = 0 which implies that or.= 1. For p = 1 (23) is satisfied by F = F(q) and G = G(q). Thus for a Newtonian fluid similarity solutions exist for all the cases under discussion here. The Newtonian value of any of the quantities plotted in these or subsequent graphs is identical to the zero-shear-rate limit of that quantity. These and all subsequent graphs were drawn by an x*y plotter interfaced with a Xerox Sigma 6 computer. The slope discontinuities in some of the curves are due to the equal spacing of points used by the plotter and not to any discontinuities in the computed solutions. For the wedge the vertex half-angle is related to p by the formula elv= TPl(l -PI.
0.2
0.4 0.6 Fig. 1. Apparent viscosity for wedge.
(42)
0.8
1.0 E
S.G.ROCHELLE and J.PEDDIESON,JR
720
0.2
0,o
0.4
0.6
0.8
E
1.0
Fig. 2. Apparent viscosity for cone.
For the cone the co~es~nding reIationsh~p is given by Rosenhead[l3]. A few representative values are as follows. Half-angles of 30“, 45”, W, and 90” correspond, respectively, to p = 0.20, l/3,0.50 and 1.0 for the wedge and to p = 0.12, l/3,0.42 and 1.0 for the cone. It is interesting to note that a similarity solution does not exist for either plane or axisymmetric stagnation-point flow 0, = 1) when the Maxwell-Oldroyd model is used. Some typical results for the shear-stress (skin-friction) coefficient
are shown in Figs. 3 and 4. It can be seen that in all cases the maximum and minimum values of the skin-friction coefficient lie at the end points of the region and that the viscoelastic skin-friction coefficient is always less than the co~espondi~ Newtonian value (often substantially less for these values of yr and ~2). This is in contrast to the case of the flat piate[lO] where an overshoot phenomenon is observed and the maximum value of the skin-friction coefficient (which is larger than the Newtonian value) occurs in the middle of the region. In observing these and subsequent figures it is important to recall that the physical distance from
0.8
0,4
0.2
Fig. 3. Shear
0.6
stress coefficient for wedge.
ha
t
1.0
721
Viscoelastic budnary-layer flows past wedges and cones
C
0.3 ’ 0.0
0.2
0.4
006
0.8 E
1.0
Fig. 4. Shear stress coefficient for cone.
the vertex is severely compressed by the use of the variable 6 (rather than s) especially for large values of s (value of 5 near unity). Thus large gradients with respect to 5 do not necessarily reflect large changes as can be seen from the relationship
where A is any variable. The variable 5 is employed only because it simplifies the numerical solution and the presentation of the data. Data for the normal-stress coefficients c,, = (2/(1 +p))W CWI
=
(2/U
+ PW1
- 5))(’- %,,(&O) = 2((r, + 72)/45,0) = 2(T*P(&O) - 5)) (’ - 3%,,(&0)
-
T3)@,ww2
(45)
73)(4mm*
for the wedges only are shown in Figs. 5 and 6. To see the significance of these quantities it is necessary to refer to (2) which, when combined with (45), yields
0.3 0.2 2.c
0.0
0.1
0.2
0.4
0.6
Fig. 5. First normal stress coeficient for wedge.
0.8
E
1.0
S. C. ROCHELLE
722
and J. PEDDlESON
JR
0.5
C nn 0.4
0.3
0,2
0.1
0.0 0 0.1 -’ Fig. 6. Second normal stress coefficient
for wedge.
&9(&O)= -(Po - R/(1 - ‘9)2”(1- F(1 + p)([/(l - %9)‘“-“c,,)/2) Ll(5.0) = -(Po- (Y(1 - 5))2p(l - &(I +p)(&l - 5))‘p-‘)Cn.)/2)
(4)
where p. is the pressure at the vertex. For a Newtonian fluid C,, = C,,, = 0. It can be seen that there is a limited region near [ = 0 where the extra normal stresses, even though multiplied by E, have a magnitude equal to or exceeding that of the pressure. It should be noted, however, that the fundamental assumption of boundary-layer theory namely a, * &a,, is violated near 5 = 0. Thus boundary-layer theory cannot be applied with confidence in this region. Away from ,$= 0, where the boundary-layer equations are applicable, the elastic normal-stress effects are small compared to the contribution of the pressure. In view of these facts it would appear that the C,, and C,, play a relatively minor role in determining the forces applied to a wedge or cone by a viscoelastic liquid. For brevity, therefore, results are shown for the wedge only. It should be recalled that the normal stress perpendicular to the s,n plane is predicted to be zero by the Maxwell-Oldroyd constitutive equation. Thus ~~~-a,,, and u”, respectively are the quantities generally referred to as the first and second normal stress differences. It can be seen that C,,-C,,, (and thus ~~,-a,,) are predicted to be positive at all wall positions. This result is in agreement with observed behavior in viscometric flows. It can be further seen that C,. (and thus Us,,) is predicted to be positive in the high-shear-rate limit and negative in the low-shear-rate limit. Experimental evidence is not conclusive on this point, with some experimenters finding the second normal stress difference to be positive in viscometric flows while others observe it to be negative. The present results suggest that the sign may be a function of the shear rate. The displacement thickness for wedge and cone flows is defined to be S=
I
a
(1 - v,Jsp) dn.
0
(47)
Figures 7 and 8 display results for the displacement-thickness coefficient C, = (( 1+ p)/2)“*(51(1 - [))‘p-1)‘2S= [( 1 - F) dn.
(48)
It can be observed that this quantity is always decreased by fluid elasticity and that for the values of yI and y2 shown the decrease can be substantial. Figures l-8 all contain results for y = 0.4. As yl approaches y2 (fixed at unity in these calculations) the apparent viscosity approaches a constant value of unity. Since the variable viscosity is the only non-Newtonian effect present in the field eqns (28), the solutions for F and G will be the same as for a Newtonian fluid when yl = y2. It should be pointed out, however, that the normal stresses do not vanish when yl = y2 unless both are equal to zero. Thus it is not
Viscoelastic
boudnary-layer
723
flows past wedges and cones
0.8
O-6
0.5
0.4 0.0
0.2
0.4 Fig. 7. Displacment
0.6
0.8
c
1.0
thickness fur wedge.
cl 3 so,4
r,=l,O
0.6
o,o
0.2
0.4 Fig. 8. Displacement
0.6 thickness for cone.
0.8
E
1
724
S. G. ROCHELLE and 1. PEDDIESON, JR 1.0
C
x = 1.0
.n
0.8
0,6
0.6
OS2
WEDGE
o,oI
I
I
I
.
*
.
I
.
0*5
0.0
.
,
E
o,ol
1,o
CONE
.
.
*
,
,
0.0
,
,
,
,
OS5
,
6
1.0
Fig. 9. Shear stress coefficient for wedge and cone.
strictly correct to refer to the situation where y1 = y2 as the Newtonian case. This terminology, however, will be used in the present discussion because, as previously stated, the normal stress effects are of secondary importance in locally viscometric boundary-layer problems. Figures 9 and 10 use the skin-friction coefficient to illustrate the amount of deviation from Newtonian behavior associated with various values of yl. It can be seen that the results for wedges and cones are qualitatively similar with about the same order of magnitude of skin-friction reduction exhibited for a given value of yI. Since the wedge and cone angles associated with a given value of p are different, a fe\N results for the skin-friction coefficient are presented in Fig. 11 for selected vertex angles. For all angles shown the results are qualitatively similar. Since constitutive equations intended to describe the behavior of real viscoelastic liquids are
ls2 C ‘”
1.2 C .” 1.0
088
L-J r,=l.O w -60”
0.6
O.l;\\
0,6
WEDGE
0.4
i 0.4
0,2 0.0
0.5
E
4 0.2 1.0 0.0
CONE
OS5
Fig. 10. Shear stress coefficient for wedge and cone.
E
1,1)
Viscoelastic boundary-layer flows past wedges and cones
0.8
0.8
086
006
12s
Fig. 11.Shear stress coefficient for wedge and cone.
highly complex it has often been proposed that approximate solutions be obtained either by perturbation methods or by using approximations to the general constitutive equations that are valid for small elastic effects. It is hoped that the solutions obtained in the present work can be used to test some of these proposed approximation methods in a specific case. Even without the aid of explicit calculations it is possible to observe that it will be easire to obtain solutions for small elastic effects when p > l/3 than when p < l/3. For the locally viscometric boundary-layer theory the approximation of small elastic effects would amount to using an approximate form of CL,found by expanding (30) for 714 1 and y2 Q 1, in (28). This would not change the parabolic nature of (28) so the solution would still have to start at ,$= 0 and proceed in the direction of increasing 6 For p > l/3 the elastic effects are smallest at 5 = 0. Thus the only question would be how close one could get to 5 = 1 before the approximation would break down. For p < l/3, on the other hand, the elastic effects are largest near ,$= 0 and expansion of (30) for y1 Q 1 and y2 & 1 would be invalid in this region because these parameters are multiplied by functions of 5 which become infinite at 5 = 0. This makes it doubtful that an approximate solution could be obtained because the initial conditions would have to be applied in a region where the approximation is invalid. If the initial conditions are not applied unavoidable ambiguities appear in the solution as indicated by the work of Davis[17] using the second-order constitutive equation. It may be possible to resolve the problem discussed above by formulating another approximate constitutive equation valid for large elastic effects and using it as a basis for solutions when p < l/3. It does not appear that much attention has yet been devoted to equations of this type. CONCLUSION
In this paper the locally viscometric boundary-layer equations associated with a MaxwellOldroyd viscoelastic fluid have been solved numerically by an implicit finite-difference method for flow of such a fluid past a wedge or a circular cone. Selected numerical results were presented graphically and the important aspects of these were discussed. REFERENCES [l] R. B. BIRD, Ann. Rev. Fluid Mech. 8, 13 (1976). [2] R. ROTHENBERGER, D. H. MCCOYand M. M. DENN, Trans. .Soc. Rheol. 17,259 (1973). [3] 2. 4. SUN and M. M. DENN, A.Z.C’.h.E.J. 18, 1010(1972). 141M. M. DENN, C. J. S. PETRIE and P. AVENAS, A.1Ch.E .l 21,791 (1975). [S] R. J. FISHER and M. M. DENN, A.Z.C.h.E .I. 23.23 (1977). 161A. C. ERINGEN, Mechanics of Conlinus, Chap. 9. Wiley, New York (1%7).
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[7] J. G. OLDROYD, Proc. Roy Sot. Lond. 245A. 278 (19.58). [8] C. -C. HSU, J. Fluid Mech. 27,445 (1967). [9] S. G. ROCHELLE and 1. PEDDIESON, Proc. Twelfth Annual Meeting Sot. Engng Sci. 1159 (1975). [IO] S. G. ROCHELLE and J. PEDDIESON, Proc. Twelfth Southeastern Seminar on Thermal Sciences 113 (1976). [li] J. PEDDIESON, Deu. Mech. 6, 153 (1971). [12] A. G. FREDRICKSON, Principles and Applications of Rheology. Chap. 7. Prentice-Hall. En&wood Cliffs (1964). [13] L. ROSENHEAD (editor), Laminar Boundary Layers. Chap. 5. Oxford University Press. London (1%3). [14] F. G. BLO’ITNER, AIAA J. 8. 193 (1970). [IS] R. E. BELLMAN and R. E. KALABA, Quasilinearization and Nonlinear Boundary-Value Problems, Chap. 2. American Elsevier, New York (1%5). 1161J. G. OLDROYD, Rheol. Acta I, 337 (]%I). [17] R. T. DAVIS, Deu. Mech. 4, 1145(1%7).
(Received
for publication
8 Juiy 1979)